Multilineare

Multilineare, or multilinear in English, refers to a property of functions that are linear in each of several input arguments separately. The concept is central to multilinear algebra and underpins the study of tensors, forms, and their transformations. A multilinear map can take multiple vector arguments and produce a scalar or another vector, while preserving linearity in every position.

Formally, let V1, V2, ..., Vk be vector spaces over a field F, and W another vector space over F. A function f: V1 × V2 × ... × Vk → W is called multilinear if, for each i from 1 to k, the map obtained by fixing all arguments except the i-th is linear in the i-th argument. Equivalently, for all vectors and scalars, f(..., a vi + b wi, ...) = a f(..., vi, ...) + b f(..., wi, ...). When W = F, f is called a multilinear form; if k = 2, f is bilinear.

Important examples include the dot product in Euclidean space, which is bilinear in its two vector arguments; the determinant function, which is multilinear in the rows (or columns) of a matrix; and the cross product, which is bilinear from R^3 × R^3 to R^3. The product of real numbers is also bilinear as a map R × R → R.

Multilinear maps relate to tensor products: the space of all k-linear maps from V^k to W is naturally isomorphic to Hom(V ⊗ ... ⊗ V, W). Variants include symmetric multilinear maps and alternating (or antisymmetric) multilinear maps, which correspond to symmetric tensors and alternating tensors, respectively.

Multilinearity is a foundational concept in areas ranging from linear algebra and tensor analysis to differential geometry and theoretical physics.